Breast cancer mortality will be drastically reduced if tumors are diagnosed and removed as early as possible before the onset of metastasis. On the basis of data from the groups working with contract-enhanced MRM, it can be expected that MRM will in the next century replace conventional mammography as the chosen procedure for breast imaging. Since the current MRM is more expensive than conventional mammography, its increased application will accelerate if a special inexpensive MRM devices appear on the market. A low cost open air self-shielded superconducting MRM has been designed and proposed. The MRM will enable the patient to be examined in the open air with surface coils very accessible for breast biopsy. The patient can be maneuvered around so that the magnet only needs to produce a smaller volume of high resolution magnetic field. Therefore the overall physical size and thus the cost of the MRM will be significantly reduced. Since the MRM is self-shielded, the cost of any facility construction, remodeling or shielding will be eliminated. This proposal will (1) design "C" shaped split gap 1.0 T self shielded superconducting MRM and (2) design a table top 0.5 T or vertical superconducting wall permanent magnets composed of high temperature superconductors. These designs will have sufficient details to allow the selection of construction of a low cost MRM in phase 2.